Among finite field arithmetic operations, the $AB^2$ operation is known as an efficient basic operation for public key cryptosystems over $GF(2^m)$,Division/Inversion is computed by performing the repetitive AB$^2$ multiplication. This paper presents two new $AB^2$algorithms and their systolic realizations in finite fields $GF(2^m)$.The proposed algorithms are based on the MSB-first scheme using standard basis representation and the proposed systolic architectures for $AB^2$ multiplication have a low hardware complexity and small latency compared to the conventional approaches. Additionally, since the proposed architectures incorporate simplicity, regularity, modularity, and pipelinability, they are well suited to VLSI implementation and can be easily applied to inversion architecture. Furthermore, these architectures will be utilized for the basic architecture of crypto-processor.